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The problem asks to match the compositions of two functions \( f(x) = 3x + 5 \) and \( g(x) = 2x^2 + 3x \) with their equivalent expressions.

We are given four options for compositions of functions:
1. \( g \circ f \)
2. \( f \circ f \)
3. \( g \circ g \)
4. \( f \circ g \)

### Let's calculate each composition:

1. **\( g \circ f(x) = g(f(x)) \)**  
   We first substitute \( f(x) = 3x + 5 \) into \( g(x) = 2x^2 + 3x \).  
   \[
   g(f(x)) = 2(3x + 5)^2 + 3(3x + 5)
   \]
   First expand \( (3x + 5)^2 = 9x^2 + 30x + 25 \):  
   \[
   g(f(x)) = 2(9x^2 + 30x + 25) + 3(3x + 5)
   \]
   Expand both terms:
   \[
   = 18x^2 + 60x + 50 + 9x + 15
   \]
   Simplifying:
   \[
   g(f(x)) = 18x^2 + 69x + 65
   \]
   This corresponds to option **D**.

2. **\( f \circ f(x) = f(f(x)) \)**  
   Substitute \( f(x) = 3x + 5 \) into itself:
   \[
   f(f(x)) = 3(3x + 5) + 5
   \]
   Expand:
   \[
   = 9x + 15 + 5 = 9x + 20
   \]
   This corresponds to option **B**.

3. **\( g \circ g(x) = g(g(x)) \)**  
   Substitute \( g(x) = 2x^2 + 3x \) into itself:
   \[
   g(g(x)) = 2(2x^2 + 3x)^2 + 3(2x^2 + 3x)
   \]
   First, expand \( (2x^2 + 3x)^2 = 4x^4 + 12x^3 + 9x^2 \):  
   \[
   g(g(x)) = 2(4x^4 + 12x^3 + 9x^2) + 3(2x^2 + 3x)
   \]
   Expand both terms:
   \[
   = 8x^4 + 24x^3 + 18x^2 + 6x^2 + 9x
   \]
   Simplifying:
   \[
   g(g(x)) = 8x^4 + 24x^3 + 24x^2 + 9x
   \]
   This corresponds to option **A**.

4. **\( f \circ g(x) = f(g(x)) \)**  
   Substitute \( g(x) = 2x^2 + 3x \) into \( f(x) = 3x + 5 \):
   \[
   f(g(x)) = 3(2x^2 + 3x) + 5
   \]
   Expand:
   \[
   = 6x^2 + 9x + 5
   \]
   This corresponds to option **C**.

### Final Answers:
1. \( g \circ f \) → **D**
2. \( f \circ f \) → **B**
3. \( g \circ g \) → **A**
4. \( f \circ g \) → **C**

Let me know if you need further explanation or have any questions!

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Here are 5 related questions for practice:
1. What is the inverse of \( f(x) = 3x + 5 \)?
2. Can you find the inverse of \( g(x) = 2x^2 + 3x \)?
3. What happens if you compose two inverse functions?
4. How do you verify if two functions are inverses?
5. What is the domain and range of the function \( g(x) = 2x^2 + 3x \)?

**Tip:** When composing functions, always remember to substitute one function into the other and carefully simplify the expression step by step!

Let f(x) = 3x + 5 and g(x) = 2x^2 + 3x. Match the compositions g(f(x)), f(f(x)), g(g(x)), and f(g(x)) with the given options.

Solve a precalculus problem involving function compositions with f(x) = 3x + 5 and g(x) = 2x^2 + 3x. Learn how to match g(f(x)), f(f(x)), g(g(x)), and f(g(x)) with the correct expressions through detailed calculations and simplification. Ideal for high school students in grades 10-12.

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